On the group of homotopy equivalences of a manifold

We consider the group of homotopy equivalences $\mathcal E(M)$ of a simply connected manifold $M$ which is part of the fundamental extension of groups due to Barcus-Barratt. We show that the kernel of this extension is always a finite group and we compute this kernel for various examples. This leads to computations of the group $\mathcal E(M)$ for special manifolds $M$, for example if $M$ is a connected sum of products $S^n\times S^m$ of spheres. In particular the group $\mathcal E(S^n\times S^n)$ is determined completely. Also the connection of $\mathcal E(M)$ with the group of isotopy classes of diffeomorphisms of $M$ is studied.